What if Gravitational Constant G Isn't?

We take this fundamental constant for granted, but determining its value with precision is surprisingly tricky -- and what if that value isn't truly constant?

Engineers and scientists live in a world defined by many metrology standards and constants. We start with time, mass, and length, and then expand to electric current, temperature, and many others. There are also fundamental physical constants such as the speed of light or Avogadro's number.

While all these constants are important, some of them are far removed from our daily lives. But one is not: the gravitational constant G. Even since Isaac Newton formulated the law of gravitational attraction F = G (mass1 × mass2)/r2, inspired by that apple falling from a tree, the value of G has been of great interest. Given how pervasive and accessible gravity is, it should be pretty easy to measure G accurately, right?

Well, yes and no. It turns out that gravity is easy to measure, but hard to measure with precision. A fascinating article in the latest issue of Physics Today, "The search for Newton’s constant," discusses the history of measuring G. It looks at the various experimental setups that have been used over several hundred years (torsion-balance, pendulum, beam-balance, and others) and the data spread in results of each. Some of the sophisticated tests by serious researchers produce results with low uncertainty, yet they differ significantly from other tests, which also claim low uncertainty.

While researchers have certainly improved the accuracy and precision of their results, the article explains why G is still so hard to measure. It's not only an interesting, well written article, it's also a sobering and thought-provoking one as well, because you likely assumed that G's value is pretty much nailed down solid, end of story.

Yet, as most engineers and scientists know, getting consistent, accurate results in any test-and-measurement challenge to better than three or four significant figures is rarely easy. Every added significant figure means ever-more-subtle sources of error must be uncovered, understood, calibrated out, or compensated for in the fixture and equipment.

If you're lucky, the test can be structured so some of these errors actually drop out, or self-cancel, much as the value of mass m cancels out in some basic physics experiments and even carnival rides, such as the "rotor ride" or Gravitron (Figure 1) where participants "stick" to the wall via centripetal force and friction. The mass of the person doesn't matter, only the size of the rotor, the speed of rotation, and the coefficient of friction between their clothes and the wall (Figure 2). (If you can't explain why the person sticks, and why their weight is not a factor, go to a basics physics book.)

Figure 1

The Gravitron's rotor spins and pins people to the wall. A functionally similar but more sophisticated version is used in centrifuges to acclimate astronauts to high-G environments. (Source: NASA)

Or maybe there's another explanation about the elusiveness of a precise, accurate value of G, one that keeps physicists and metrologists worrying: Perhaps the "squared" exponent in the denominator of Newton's Law is not exactly 2.0 out to as many places as you care to pick. Or maybe G itself is not a true constant, but actually changes slightly over time and place. Stranger things have happened; just ask those physicists who believed in the absoluteness of time and distance, but had to change their beliefs to accommodate the curvature of time and space, as well as time dilation itself and even E = mc², as Einstein's 1905 paper on Special Relativity became accepted principle.

Figure 2

The rotor ride spins and people inside the cylinder stick to the wall, irrespective of their mass. Riders are subject to three forces: weight, normal force, and frictional force.
(Source: stuegli.com)

Have you ever had a constant or fixed assumption in engineering or science that you had to abandon or at least become flexible about? Have you ever stopped and wondered what "gravity" is, as well? What are your thoughts are gravity waves and gravitational frame-dragging, as Gravity Probe B is exploring? (See "Spinning spheres test relativity's subtlety" and "The Gravity Probe B Bailout.")

An alternative to measuring sea surface height to extreme precision, which would require knowing the satellite height to extreme precision along with all the other phenomena affected the RF signal propagation, is to measure the sea height relative to fixed objects.

For example, if the local sea height is measured with respect to the nearly simultaneously measured land height, then it could be easier to infer variations in the local sea height.

Hmmmm...that assumes gravity propagates like light--but we really don't understand much about gravity. Still looking for gravity waves, and trying to understand gravity as part of the fabric of space. So I don't know if your assumption about "conservation" nor your analogy to light apply here. Great minda are working on the problem!

With respect to 'erhaps the "squared" exponent in the denominator of Newton's Law is not exactly 2.0,' this exponent usually occurs because there is conservation in an expanding sphere. For example, a pulse of light emitted from a point source and spreading uniformly in all directions is effectively spreading on the surface of an expanding sphere. Since the surface area of this sphere is proportional to r^2.000000..., we have a 1/r^2 law for the intensity of light.

Land-based lasers are used to measure the altitude of the satellites which use RADAR to measure the sea surface height. But the land-based lasers are themselves experiencing vertical movement due to the solid earth tides caused by lunar and solar grivity. They may also be experiencing long-term horizontal and vertical movement due to tectonic plate movement. I have no doubt that all of these factors, and plenty more, are build into the model that is used to calibrate the system and to correct the raw data. I think NASA is probably the most trusted part of the US government, and rightly so. But NASA also has to constantly worry about funding, so it may be difficult for administrators to admit the limitations of their systems. (e.g. warnings of engineers not taken seriously by administrators prior to Challenger disaster) There may be a temptation for NASA administrators and NASA press correspondents to use "typical" values instead of "worst case" values.

A lot of the literature I have read talks about "precision", but precision is not the same thing as accuracy.

The satellites Jason-1 and Jason-2 are part of the much larger "geodetic infrastructure". The National Research Council issued a report in 2010 entitled "Precise Geodetic Infrastructure - National Requirements for a Shared Resource". I bought a copy online, and it is fascinating reading. But the following statement gave me pause: "Modern geodesy delivers precision to one part per billion, and precision of one part per trillion can be envisioned in the foreseeable future.".

Presision, maybe, but I can't help wondering about accuracy, which depends upon having a stable reference point. If every part of the earth is moving, then all calibration must be performed with respect to an imaginary reference point, such as the earth's center of mass. After all of the available data is factored into the model, aren't we still left with a lot of assumptions?

@Bob: And the moon's gravity causes not only ocean tides, but also the so-called "solid earth tide" which causes the land under your feet to rise and fall tens of centimeters every 12 hours.

Keeping this in mind, it's amazing to me that we don't have more Earthquakes than we do. So if we went ~4 billion years back in time when the Moon had just formed and was much closer to the Earth than it is today, how big would the sea and earth tides have been then?

@Bill: It's naive and irresponsible, IMO, to think the Sun's output has been constant over the years...

Surely we know it's not been constant. Did you see my book review of Alone in the Universe by John Gribbin. People talk about the Earth inhabiting the "Goldilocks Band" that's warm enough to have liquid (none-ice) water yet cool enough that the water doesn't boil off. As I recall, Gribbin presented lots of evidence for variations in the sun's temperature/outout over time showing the the Goldilocks Band moved in or out over time and that Earth was lucky to have always remained within the extreme end points.